00-195 G.M. Molchan
Maximum of fractional Brownian motion: probabilities of small values (43K, LATeX) Apr 21, 00
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Abstract. Let $b_\gamma (t)$, $b_\gamma(0)=0$ be fractional Brownian motion, i.e., a Gaussian process with the structure function $E|b_\gamma (t) - b_\gamma (s)|^2 = |t-s|^\gamma$, $0<\gamma<2$. We study the logarithmic asymptotics of $P_T = P\{b_\gamma (t) < 1,\quad t \in T\Delta \}$ as $T \to \infty$, where $\Delta$ is either the interval $(0,1)$ or a bounded region that contains a vicinity of $0$ for the case of multidimensional time. It is shown that $\log\,P_T = -D\log\,T(1+o(1))$, where $D$ is the dimension of zeroes of $b_\gamma(t)$ in the former case and the dimension of time in the latter.

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